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# 2011 Calculus AB free response #4d

Mean Value Theorem and differentiability. Created by Sal Khan.

## Want to join the conversation?

• Why is the reason "not differentiable " rather than "not contionous" Is there really a difference on the AP Exam • So does the Mean Value Theorem not apply to any function that is at some point not differentiable, or only if that point is x=0? • At , why does Sal multiply the integral by 1/7? Should it not have been -2/7 instead?
(1 vote) • Because the average value for integrals states: f(c)=1/(b-a)•∫[a,b] f(x)dx, all one has to do is plug in numbers. Letting a= -4 and b= 3 when you place these numbers into f(c) you will get f(c)= 1/(7)•∫[-4,3] f(x)dx. At this point the math is straight forward. Now note that on the inside of the definite integral (integrand) are the slopes of f(x) everywhere. So when integrating the derivative function you come back to the original function while also having the capability of getting the area underneath the curve. Since the average value, f(c), is equal to one over the two endpoints of f(x) (the original function) subtracted from each other, multiplied by the definite integral (the area of f(x)), generates the average rate of change of the entire graph of f(x). Hopefully I didn't confuse you with all the extra material behind reasoning.
• When calculating the average rate of change, why does Sal use average value? I know these two are not the same so why did he used the average value equation (1/b-a)(integral from a to b of f(x)dx) to find the rate of change?
(1 vote) • Okay, so the question gives us this statement, "There is no point c, -4<c<3, for which f '(c) is equal to that average rate of change." This basically means: 'The MVT doesn't apply.' And we know that this is because f is not continuous or differentiable at x = 0. So, the statement is true.
The question asks us to tell WHY the (true) statement does NOT contradict the MVT?
So does I read that question as saying why "this statement" does not DISprove the MVT. Which means say why it AGREES w/ what the MVT says (not agrees as in applies).

...Am I right?

This isn't so much as a mathematical/calculus problem as it is a question about reading the AP question. Sorry if I broke the rules. Thanks!
(1 vote) • I'm not sure why the statement does not contradict the Mean Value Theorem. Sal had me at the actual mathy portion of the problem but lost me from that point forward.
(1 vote) • At , Sal says that there would be a point where the derivative equaled the average slope (i.e. the Mean Value Theorem would apply). However, wouldn't the MVT still not apply since the derivative of f doesn't exist when x = -3 (because the slope of that point is undefined)? 